Random walks with heavy tails and limit theorems for branching processes with migration and immigration (Q2888812)

The authors consider two types of branching models that are designed to reflect two features of real-world populations which are not reflected by many classical branching processes. The first feature is stationarity of the total population size (in contrast to explosion-extinction principles that many branching processes exhibit). The second feature concerns the distribution of the population in space, namely, the characteristic of having large areas of mostly empty space and small areas in which many individuals are accumulating.NEWLINENEWLINEThe first model is a discrete-space model with populations residing at each lattice point \(x \in \mathbb{Z}^d\). At each time \(t\) the total population at \(x\) is denoted by \(n(t,x)\), \(x \in \mathbb{Z}^d\). The initial populations \(n(0,x)\), \(x \in \mathbb{Z}^d\), are assumed to be i.i.d.\ with Poisson distributions. Particles die independently of each other at rate \(\mu>0\) and produce exactly one offspring (or split) at rate \(\beta > 0\). The model further includes a migration mechanism that allows particles to make (heavy-tailed distance) jumps to other sites and an immigration mechanism that allows immigration of new particles at overall rate \(\kappa > 0\). The branching process is critical iff \(\mu = \beta + \kappa\).NEWLINENEWLINEThe second model is the continuous contact model in \(\mathbb{R}^d\) introduced by \textit{Yu. Kondratiev} and \textit{A. Skorokhod} [``On contact processes in continuum'', Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, No. 2, 187--198 (2006; Zbl 1096.60040)].NEWLINENEWLINEIn the paper, the authors develop the moment theory for the first model in the critical case and in dimension \(d \leq 2\). Further, they derive the asymptotic variance of particles in a region as the region grows larger. For the second model, the authors establish the tightness of the finite-dimensional distributions of the process.